← Moser's Worm Problem / Rounds / Round 11
Round 10's 18 interval boundaries were still described mainly by double-precision numerical root-finding. This round makes three advances: it rewrites all contact boundaries as exact closed-form expressions (in terms of fractions of α, β, π, differing from Round 10's numerical boundaries by at most 5.24×10⁻¹⁴); it builds adaptive derivative boxes for all 18 intervals — 579 sub-boxes in total, 0 unresolved — proving that a zero derivative sign is excluded everywhere in the interior of the intervals; and it builds an independent closed-form error envelope for the most sensitive branches, 120° and 270°, giving the first explicit box: s₁₂₀-s₂₇₀∈[1.6350×10⁻⁹,1.6417×10⁻⁹], with the lower bound strictly positive. All 12 smooth stationary-point root boxes also pass second-derivative zero-exclusion. Honest boundary: NumPy's basic operations are not directed-rounding throughout, so the precise characterization is a “semi-verified computational certificate with closed-form error bounds, floating-point outward rounding, and box-by-box exclusion” — substantially stronger than grid scanning, but still short of the rigorous certificates from dedicated interval libraries such as Arb and MPFI.
Relationship to other packages, stated as far as possible in the document's own words, not my interpretation.
“This is the first time in this series that an explicit error box has been given for the 10⁻⁹-level 120°/270° competition, rather than simply comparing two high-precision central values.” — from Section 3, “Closed-Form Envelope for the Special Branches,” of this package's main document.
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